Grid transparency and grid hole pattern control for ion beam uniformity

ABSTRACT

A design process for varying hole locations or sizes or both in an ion beam grid includes identifying a control grid to be modified; obtaining a change factor for the grid pattern; and using the change factor to generate a new grid pattern. The change factor is one or both of a hole location change factor or a hole diameter change factor. Also included is an ion beam grid having the characteristic of hole locations or sizes or both defined by a change factor modification of control grid hole locations or sizes or both.

RELATED APPLICATIONS

This application claims benefit of U.S. Provisional Application No.60/677,386,entitled “GRID TRANSPARENCY AND GRID HOLE PATTERN CONTROL FORION BEAM UNIFORMITY” and filed Mar. 31, 2005, specifically incorporatedby reference herein for all that it discloses or teaches. The presentapplication is also a continuation of U.S. patent application Ser. No.11/395,354,entitled “GRID TRANSPARENCY AND GRID HOLE PATTERN CONTROL FORION BEAM UNIFORMITY” and filed Mar. 31, 2006, specifically incorporatedby reference herein for all that it discloses or teaches.

TECHNICAL FIELD

The described subject matter relates to a technique of ion source gridhole pattern design and control of grid transparency using stretchingand/or shrinking of distances between grid holes, typically radially orlinearly, where the grids are typically the electrodes of a broad beamion acceleration system.

BACKGROUND

One of the issues in the development and usage of broad beam ion sourcesis in the production of very uniform ion beam density profiles. Sinceelectric discharge plasmas do not themselves have a uniform distributionof ion density, ion sources which utilize ions generated in the plasmatypically have non-uniform ion beam density profiles. Though thediscussion here references beam grids, for example for ion sources, itapplies generally to any charged particle broad beam source includingboth positive and negative ion beam sources and electron sources.

In order to solve this problem and achieve higher uniformities of ionbeam current densities, ion extraction grids of gridded ion sources havebeen developed with variations of grid open area fraction (gridtransparency) over the entire grid pattern. Indeed many have discretesections of grid patterns in each of which may be different hole-to-holedistances and/or different hole diameters. Such ion extraction gridshave been provided as solutions for various applications. At higherrequirements of ion beam current density uniformities, however, theboundaries of such discrete sections may still cause unwanteddisturbances in ion beam current density uniformities.

FIG. 1 shows one example of an arbitrary conventional grid pattern whichhas multiple discrete radially defined zones for grid transparency (theconcentric circles being indicative of, and demarcating the boundariesbetween zones). Within each zone, the grid design is typically filledwith repeating patterns to obtain a constant grid transparency withinthat zone. At the boundaries of zones, however, the transition betweenone zone and another may not be smooth, resulting in localdiscontinuities in the grid hole density, and, if left uncorrected, inthe beam current density. FIG. 2 shows typical irregularities at bothradial and azimuthal zone boundaries (note, the small black circles showlocations of holes if they were equally spaced). These zone boundariesmay be radial and various azimuthal boundaries may appear as shown hereor, depending on the particular design, there may be other boundarieswhere either the hole size or spacing changes discontinuously. (Note,the six areas unpopulated with holes that are observed at the secondradial boundary from the center in FIG. 1 are a result of other designfeatures not relevant to the subject of this disclosure.)Conventionally, any adjustment to smooth the transition at theboundaries has been done by design personnel on a hole-by-hole basis.Shown in FIG. 3 is a distribution of grid transparency using anarbitrary unit as a function of radius in another arbitrary conventionalgrid design. The scattered data points in FIG. 3 are associated withboundaries where the patterns do not match and holes have been manuallyadjusted.

SUMMARY

Disclosed is a design process for varying hole locations or sizes in anion beam grid including identifying a control grid to be modified;obtaining a hole location and/or hole size change factor for the gridpattern; and, using the change factor to generate a new grid pattern,which may also be referred to as “scaling” the grid transparency.Further disclosed are grids generated using the described designpatterns.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings:

FIG. 1 is a schematic plan view of a conventional ion beam grid pattern;

FIG. 2 is an enlarged portion of the schematic grid pattern of FIG. 1;

FIG. 3 is a plot of the distribution of transparency of anotherconventional grid pattern;

FIG. 4 presents a pair of enlarged portions of a schematic grid pattern;

FIG. 5 presents a pair of enlarged portions of an alternative schematicgrid pattern;

FIG. 6, which includes sub-part FIGS. 6A, 6B, 6C and 6D, is a collectionof graphs illustrating a process hereof;

FIG. 7 is another schematic plan view of a conventional ion beam gridpattern;

FIG. 8 is a schematic plan view of an ion beam grid pattern generatedhereby;

FIG. 9 is a plot of a radial plasma ion density profile which may beused herewith;

FIG. 10 is a flow chart depiction of a design process hereof; and,

FIG. 11 is another flow chart depiction of a design process hereof.

DETAILED DESCRIPTION

An objective design process for varying hole locations and thehole-to-hole distances therebetween in an ion beam grid has beendeveloped using a mathematical model to change grid transparency, alsoknown as grid open area fraction. The new technique or techniques hereofdo not require discrete zones (such as those shown in FIGS. 1 and 2)and, therefore, may substantially eliminate tedious and subjectivemanipulation of moving grid holes at boundaries of the zones. Rather,this design process provides for substantially continuously varying by“stretching” and/or “shrinking” the hole-to-hole distances tocontinuously or substantially continuously vary the grid iontransparency (open area) to compensate for plasma non-uniformities orbeam non-uniformities due to other causes. In addition to the process,also thus developed are new grids manufactured with a substantiallycontinuously varying transparency based on a design process hereof. Theresult of either or both will typically be greater control of ion beamuniformity. Though the discussion here references beam grids, forexample for ion sources, it applies generally to any charged particlebroad beam source including either positive or negative ion beam sourcesor electron sources.

This mathematically based process for developing grid transparencies maysolve the shortcomings of current grid pattern design techniques, whichuse discrete sections or zones of established hole-to-hole distances orhole diameters, by instead using a substantially smooth, substantiallycontinuous variation of hole-to-hole distances or substantiallycontinuously changing grid hole diameters. Exemplar design proceduresaccording hereto are thus described in the following procedures.

A first general technique starts with an initial (original or control)grid pattern to be modified. Then, in a first alternativeimplementation, as shown in FIG. 4, radial coordinates of the initialpattern hole locations may be changed from r to R. However, when doingso, there should be the same number of grid holes in the range of(r−dr/2, r+dr/2) in the original pattern and in the range of (R−dR/2,R+dR/2) in the modified pattern. Assuming the grid transparency of theoriginal and modified grid patterns may be given as functions of radiusas f(r) and F(R), respectively,

${{{f(r)} \times 2}\pi\;{r \times {\mathbb{d}r}}} = {{{{{F(R)} \times 2}\pi\;{R \times {\mathbb{d}R}}}\therefore\frac{\mathbb{d}R}{\mathbb{d}r}} = {\frac{r \times {f(r)}}{R \times {F(R)}}.}}$By solving this differential equation for the grid hole location changefactor R(r) with a boundary condition of R(r=r₀)=r₀, a new grid patterncan be obtained which achieves a desired or desirable grid transparencyprofile F(R). Hereafter, the grid hole location change factor mayalternatively be referred to as a location change factor or simply as achange factor.

In an alternative second implementation, the technique may also startwith an initial (original or control) grid pattern to be modified. Then,as shown in FIG. 5, one of the Cartesian coordinates of each of severalinitial pattern hole locations may be changed from x to X, and asbefore, there should be the same number of grid holes in the range of(x−dx/2, x+dx/2) in the original pattern and in the range of (X−dX/2,X+dX/2) in the modified pattern. Assuming the grid transparency of theoriginal and modified grid patterns are given as functions of Cartesiandistance as h(x) and H(X), respectively,

${{h(x)} \times {\mathbb{d}x}} = {{{{H(X)} \times {\mathbb{d}X}}\therefore\frac{\mathbb{d}X}{\mathbb{d}x}} = {\frac{h(x)}{H(X)}.}}$Then, by solving this differential equation for a location change factorX(x) with a arbitrary boundary condition (such as X(x)=0), a new gridpattern can be obtained, which achieves the desired grid transparencyprofile H(X). An example of an application of the general process hereofis described in relation to and shown in FIG. 10 (see description below)in flow chart form for both radial and linear pattern modifications.Though not described in detail here, a similar process could be used forazimuthal pattern modifications, where ⊖ is the azimuthal coordinate ina polar coordinate system and a comparable differential equation issolved for a location change factor ⊖(θ) with appropriate boundaryconditions.

Various means may be used in solving differential equations such asthese, as for example, may be found in commercially available softwareproducts such as MathCad or Mathematica. As another example, MicrosoftExcel files have also been used for this purpose. A fourth (4^(th))order Runge-Kutta routine is one example that may be used for solutionof the differential equation, among other forward-marching techniques ofany orders (such as the so-called Euler method if it is 1st order,e.g.). For the radial method, by providing f(r) and F(R), the user wouldthen be able to obtain a location change factor R(r), which is used toradially move the grid hole locations from the original design positionto obtain the modified hole pattern. Similarly, by providing h(x) andH(X), the user would then be able to obtain a location change factorX(x), which is used to linearly move the grid hole locations from theoriginal design position to obtain the modified hole pattern for linearstretching/shrinking in Cartesian coordinate systems. Similarly, byproviding p(θ) and P(⊖), the user would then be able to obtain alocation change factor ⊖(θ), which is used to azimuthally move the gridhole locations from the original design position to obtain the modifiedhole pattern in polar coordinate systems.

Viewed in a different light, a procedure hereof may be described asfollows using a ratio of grid transparency modification as anapproximation. Step 0: Defining a desired ratio of grid transparency bythis modification (i(x) in a Cartesian system or g(r) in a 2D radialsystem) to achieve a better ion beam density uniformity profile. Suchprofiles can be determined based on experiments or other methods. Thismay first involve identifying a control grid which may be sought to bemodified. Then, Step 1: Based on the desired modification, a gridtransparency change ratio may be approximated by solving differentialequations, which have different formats depending on the direction ofmodification. In a Cartesian system, it is

${\frac{\mathbb{d}x^{\prime}}{\mathbb{d}x} = \frac{1}{{\mathbb{i}}(x)}},$where the grid transparency change ratio i(x)=H(x)/h(x) is anapproximation to H(X)/h(x) and x′(x) yields an approximation of locationchange factor X(x). In a 2D radial system, it is

${\frac{\mathbb{d}r^{\prime}}{\mathbb{d}r} = \frac{r}{r^{\prime} \times {g(r)}}},$where the grid transparency change ratio g(r)=F(r)/f(r) is anapproximation to F(R)/f(r) and r′(r) yields an approximation of locationchange factor R(r). Next, Step 2: Applying the solution of the abovedifferential equation to the control grid hole pattern, a new gridpattern is obtained. As an optional further step, Step 3: If ion beamdensity profiles obtained with the new grid pattern do not provide thepreferred level of uniformity, then, the above steps 1-2 (eitheralternatively or in addition to steps 0-2) with adjustment of thedesired modification in grid transparency may be iterated. A flow ofiterative steps according hereto is shown in and described relative toFIG. 11 (see description below). Again, a similar process could be usedfor azimuthal pattern modification.

This can be depicted somewhat graphically as shown in the four parts ofFIG. 6 using a radial example. The first graph of FIG. 6, namely FIG.6A, shows a non-uniform ion beam density profile as such might appear toa substrate of an approximate ten inch width. Note the mirroring left toright (about a central vertical axis) as might be expected from acircular grid as shown for example in FIG. 1. Note also that as might beexpected, the density generally declines near the far edges, although ahigher than normalized density may occur near or nearer the center. Incontrast, the preferred uniform normalized density should morepreferably appear as a flat line such as that shown in the third graphof FIG. 6, namely, FIG. 6C. Thus, the first step as set forth herein isto identify the grid wanting an improvement toward uniformity. The gridwhich produced the profile of FIG. 6A may be just such a grid. Then, thenext step is to calculate the desired grid transparency change ratio.This change ratio is approximated by calculating a ratio between thedesired and actual normalized beam current density at each radialposition. The second plot of FIG. 6, namely, FIG. 6B, presents agraphical representation of such a grid transparency change ratio forthe right half of the FIG. 6A graphical profile. Note, the top to bottomhorizontal mirror image effect of the FIG. 6B plot relative to the rightside portion of the FIG. 6A graph. Conceptually, the grid transparencychange ratio plot of FIG. 6B may also represent an inverse relationshipto the values of the initial values of the FIG. 6A graph. As such, theproduct of the initial values with the inverse values would approach ifnot equal the normalized value of the FIG. 6C graph. The differentialequation solution r′(r) for this radial example is shown graphically asa difference of r′(r) from r in FIG. 6D. It represents a continuouslyvarying location change factor that can be applied to the radiallocation of each hole in the original design to provide a new design,which will have improved uniformity performance approaching the idealshown in FIG. 6C. Note that, while the holes at the edge movesignificantly (up to in this particular example about 2mm), adjacentholes also move a similar amount so that the distance between adjacentholes changes only slightly. Therefore, in the example discussed above,the radial location of every hole in the original discrete zoned patternwas moved, i.e., “stretched/shrunk,” in a manner such that the newhole-to-hole spacing continuously or substantially continuously variedfrom the original spacing, resulting in a new pattern that, thoughperhaps still containing discrete zones, may better satisfy the improveduniformity criteria.

In addition to using the technique for improving the performance of aninitial discrete zoned pattern, the technique can also be used fordesign of a new pattern with continuously varying hole spacing. In oneof the examples for this case, a continuously varying grid transparencychange ratio g(r) or i(x) may be used to approximate a discrete zoneddesign and input to the differential equation. The resulting locationchange factor solution r′(r) (approximation of R(r)) or x′(x)(approximation of X(x)) may functionally duplicate yet “smooth” theoriginal discrete zoned design. A sample of such a discrete zonedpattern is shown in FIG. 7. In FIG. 7, the irregularity of the holepattern can easily be seen in areas extending radially outwardly fromthe six areas of absent holes. By application of the process describedhere, a smooth, continuously varying adjustment of grid hole locationscan then provide the smooth pattern of the grid as shown for example inFIG. 8. The result may then be greater control of ion beam uniformitybecause the discontinuities associated with the discrete zones may thushave been eliminated.

Thus, a mathematical model was developed to continuously orsubstantially continuously scale a grid pattern design to achieve adesired improved ion beam uniformity. A few grid sets were designedusing this model and etch rate measurements showed improvement of ionbeam uniformity when they were used in ion beam etch applications. Thetechnique has been applied to ion source grid design, and tests indicatepredictable improvement of grid performance in terms of ion beam currentdensity uniformity, and improved etch rate distributions. Note that insome cases the initial designs were discrete zoned control designs whichwere “stretched” continuously to compensate for non-uniformities foundin experimentally measured etch profiles. Moreover, this design processhas been iterated in series with a ray-tracing model to provide acontinuously varying transparency design. This iteration process wasused to functionally duplicate yet “smooth” the original discrete zoneddesign. The intent may thus be to provide a continuously varyingtransparency design without any discrete zones but that wouldfunctionally duplicate the original discrete zoned design as closely aspossible with still greater etch uniformity. This continuously varyingbaseline design may then serve as a starting point for furtherexperimentally based iterations to optimize grid performance at specificoperating conditions relevant to a particular application.

It has thus been found that in general, continuously or substantiallycontinuously varying transparency designs may be desirable. Given ameasured plasma density radial profile or a radial beam current densityas a starting point, this technique is capable of providing suchdesigns.

Yet another implementation of varying grid transparency designs usingchange factors such as described herein may be to substantiallycontinuously change hole diameters. A process for establishing a designwith substantially continuously varying hole diameters may beimplemented in the following manner. When an original grid design havinghole diameters, d, as a function of their radial coordinates, r, asd(r), is established and a desired modified grid design of a desiredgrid transparency with a grid transparency change ratio j(r) is known;then, a modified grid design should have hole diameters of:d′(r)=d(r)×√{square root over (j(r))}where d′(r) is a new diameter, which is designated hereafter as a holediameter change factor or, alternatively, may simply be referred to as achange factor. Alternatively, if the hole diameters and desired gridtransparency change ratio are given in a Cartesian system as d(x) andk(x), respectively, then the modified grid design diameters would beshown by:d′(x)=d(x)×√{square root over (k(x))}where d′(x) is a new diameter, which is also designated hereafter as ahole diameter change factor or simply as a change factor.

Moreover, techniques of varying hole positions and hole diameters mayfeasibly be combined together. An overall desired grid transparencychange ratio, l(x) or m(r), can be achieved by combining a desired gridtransparency change ratio, k(x) or j(r), which may be achieved byvarying hole diameters and a different desired grid transparency changeratio, i(x) or g(r), which may be achieved by varying hole locationswhere:l(x)=i(x)×k(x),orm(r)=g(r)×j(r).

For each overall change ratio, l(x) or m(r), a designer may arbitrarilypartition grid transparency change ratios (i(x) and k(x), or g(r) andj(r)) as long as the above equations are satisfied. Then, each gridtransparency change ratio can be used to solve for a location changefactor, x′(x) or r′(r), and a diameter change factor, d′(x) or d′(r).

While the various implementations described above used as a startingpoint an original or control grid pattern together with thecorresponding measured beam current density profile from the extractedbeam to determine a grid transparency change ratio for a new grid, it isalso possible to start with either a measured or a theoretically modeledplasma ion density profile. This could be useful for designing the firstiteration grids for a new device based on either a theoretical model forthe discharge or measurements of ion plasma density in a prototypechamber. Here the distinction is made between the plasma ion densityprofile associated with the ion source that supplies ions to the gridsystem and the density profile of the ion beam extracted by the gridsfrom that source. An example of such a radial ion density profile isshown in FIG. 9. In this case, one might begin the process using as acontrol grid pattern one with a constant hole spacing and use as f(r)the function representing the plasma ion density profile or any otherinitial starting point.

A further detailed process herefor may be as follows, and as shown inFIG. 10. In particular, a first step may generally involve theestablishment of a control grid pattern whether of a previouslycalculated or previously manufactured grid, which grid transparency isgiven as f(r) or h(x). Then, after the designer specifies a new gridtransparency F(R) or H(X), the grid hole location change factor may beobtained by solving one or more differential equations. An example ofsolving the differential equation(s) may include a usage of a 4th orderRunge Kutta method. For example, when a radial differential equation of

$\frac{\mathbb{d}R}{\mathbb{d}r} = \frac{r \times {f(r)}}{R \times {F(R)}}$is solved with this method where locations and boundary conditions(r_(n) (n=0,1,2, . . . ) and R₀=r₀) are defined, inter alia (forexample, various design constraints such as grid pattern overall sizeand/or minimum thickness between holes), R_(n+1) would be obtained usingr_(n), R_(n), f(r), F(R), and Δr(=r_(n+1)−r_(n)) according to thefollowing equations:

${k_{1} = {\Delta\;{r \times \frac{r_{n} \times {f\left( r_{n} \right)}}{R_{n} \times {F\left( R_{n} \right)}}}}},{k_{2} = {\Delta\;{r \times \frac{\left( {r_{n} + \frac{\Delta\; r}{2}} \right) \times {f\left( {r_{n} + \frac{\Delta\; r}{2}} \right)}}{\left( {R_{n} + \frac{k_{1}}{2}} \right) \times {F\left( {R_{n} + \frac{k_{1}}{2}} \right)}}}}},{k_{3} = {\Delta\;{r \times \frac{\left( {r_{n} + \frac{\Delta\; r}{2}} \right) \times {f\left( {r_{n} + \frac{\Delta\; r}{2}} \right)}}{\left( {R_{n} + \frac{k_{2}}{2}} \right) \times {F\left( {R_{n} + \frac{k_{2}}{2}} \right)}}}}},{k_{4} = {\Delta\;{r \times \frac{\left( {r_{n} + {\Delta\; r}} \right) \times {f\left( {r_{n} + {\Delta\; r}} \right)}}{\left( {R_{n} + k_{3}} \right) \times {F\left( {R_{n} + k_{3}} \right)}}}}},{R_{n + 1} = {R_{n} + {\frac{k_{1} + {k_{2} \times 2} + {k_{3} \times 2} + k_{4}}{6}.}}}$The solution to the differential equation may then give the new locationfor the hole as a grid hole location change factor in the form of arraysof r_(n) and R_(n). Note, tables or other utilities may be used fordeveloping and/or tracking the modified locations relative to theoriginal locations. This method can be used for radial and/or linearcoordinate systems as indicated in FIG. 10.

The alternate implementation shown in FIG. 11 start with approximatedgrid transparency change ratio, g(r) or i(x). In this implementationexample, the initial operation or Step 0, may involve defining a desiredratio of grid transparencies as g(r) (which may be obtained asF(R=r)/f(r)) or i(x) (which may be obtained as H(X=x)/h(x)). Such anexample is shown in FIG. 6B. Then, in a further operation, here, Step 1,a combination of a definition and calculation with a subsequentdetermination may be performed. First, this may involve a defining ofr_(n) or x_(n) (n=0,1,2, . . . ) with a setting of r′₀=r₀ or x′₀=x₀ andsolving one or both of the following differential equations:

${\frac{\mathbb{d}r^{\prime}}{\mathbb{d}r} = \frac{r}{R \times {g(r)}}},{or}$$\frac{\mathbb{d}x^{\prime}}{\mathbb{d}x} = \frac{1}{{\mathbb{i}}(x)}$which gives a grid hole location change factor in the form of arrays ofr_(n) and r′_(n) or that of x_(n) and x′_(n). Then, proceeding to Step2, a grid may be manufactured with a hole pattern based on the relationsof r_(n) and r′_(n) or of x_(n) and x′_(n). After this Step 2, then afurther determination of whether the new design yields a desired gridtransparency can be made, where if so the process of FIG. 11 iscomplete. However, if not, then an iteration of both Steps 1 and 2 maybe re-done. If necessary, a modification of a desired grid transparencychange ratio, g(r) or i(x), based on a grid transparency may be obtainedrelative to and/or from a previous iteration.

Though many of the examples above mention ion beam grids and ion beamsources, the design processes and grids manufactured based thereon couldapply generally to any charged particle broad beam source includingeither positive or negative ion beam sources or electron sources. Insuch cases, the spatial functions of interest would be for example thosesuch as the upstream negative ion density (and/or arrival rate) profilesor electron density (and/or arrival rate) profiles and the correspondingdownstream charged particle beam current density profiles, whereupstream and downstream are defined relative to the extraction grid.Likewise, though various forms of electrical discharge sources, forexample DC or RF excited discharges, are common as plasma sources forion beam extraction, the design processes described here and the gridsmanufactured thereon are expected to be generally applicable to chargedparticle beams extracted from plasmas generated by any one of numerousalternate means, for example, microwave plasmas, standing wave sheetplasmas, laser stimulated plasmas, surface contact or emission plasmas,and from various non-plasma, single charge species field and surfaceemission devices, inter alia.

The above specification, examples and data provide a description of theprocess and structure and use of exemplary embodiments of the invention.However, other implementations are also contemplated within the scope ofthe present invention, including without limitation methods of providingand/or grids having holes of different shapes, sizes, and locations thanthose shown and/or described. In addition, while the description hasdescribed exemplary process and grids, other processes and grids may beemployed within the scope of the invention. Since many implementationscan be made and/or used without departing from the spirit and scope ofthe invention, the invention resides in the claims hereinafter appended.

1. A beam grid comprising: a pattern of beam grid holes with a regionhaving a grid hole density that continuously varies with distance from areference on the beam grid.
 2. The beam grid of claim 1, wherein thepattern of beam grid holes is derived from an initial pattern of beamgrid holes using a change factor.
 3. The beam grid of claim 2, whereinthe change factor is applied iteratively to the initial pattern of beamgrid holes to obtain the pattern of beam grid holes.
 4. The beam grid ofclaim 1, wherein the reference is a point within a radial coordinatesystem and the grid hole density of the region varies radially from thepoint.
 5. The beam grid of claim 1, wherein the reference is at a centerpoint of the beam grid.
 6. The beam grid of claim 1, wherein the beamgrid is generally circular.
 7. The beam grid of claim 1, wherein theregion having a continuously varying grid hole density includes all ofthe pattern of beam grid holes.
 8. The beam grid of claim 1, wherein thepattern of beam grid holes has no discrete regions with different gridhole densities.
 9. The beam grid of claim 1, further comprising: apattern of mounting holes arranged about a perimeter of the beam gridfor mounting the beam grid to an ion source.
 10. The beam grid of claim1, wherein the region has a constant beam grid hole size.
 11. The beamgrid of claim 1, wherein the pattern of beam grid holes is derived froman initial pattern of beam grid holes.
 12. The beam grid of claim 1,wherein the reference is a point within a Cartesian coordinate systemand the grid hole density of the region varies linearly from the point.13. The beam grid of claim 1, wherein the region has a continuouslyvarying transparency.
 14. A beam grid comprising: a pattern of beam gridholes with region having a grid hole density and grid hole size thatcontinuously varies with distance from a reference on the beam grid. 15.The beam grid of claim 14, wherein the pattern of beam grid holes isderived from an initial pattern of beam grid holes using a changefactor.
 16. The beam grid of claim 15, wherein the change factor isapplied iteratively to the initial pattern of beam grid holes to obtainthe pattern of beam grid holes.
 17. The beam grid of claim 14, whereinthe reference is a point within a radial coordinate system and one orboth of the grid hole density and the grid hole size of the regionvaries radially from the point.
 18. The beam grid of claim 14, whereinthe reference is a center point of the beam grid.
 19. The beam grid ofclaim 14, wherein the beam grid is generally circular.
 20. The beam gridof claim 14, wherein the region having a continuously varying grid holedensity and grid hole size includes all of the pattern of beam gridholes.
 21. The beam grid of claim 14, wherein the pattern of beam gridholes has no discrete regions with different grid hole sizes.
 22. Thebeam grid of claim 14, wherein the pattern of beam grid holes has nodiscrete regions with different grid hole densities.
 23. The beam gridof claim 14, further comprising: a pattern of mounting holes arrangedabout a perimeter of the beam grid for mounting the beam grid to an ionsource.
 24. The beam grid of claim 14, wherein the pattern of beam gridholes is derived from an initial pattern of beam grid holes.
 25. Thebeam grid of claim 14, wherein the reference is a point within aCartesian coordinate system and one or both of the grid hole density andthe grid hole size of the region varies linearly from the point.
 26. Thebeam grid of claim 14, wherein the region has a continuously varyingtransparency.
 27. A beam grid comprising: a modified pattern of beamgrid holes with a region having a grid hole density that continuouslyvaries with radial distance from a reference point on the beam grid anda constant grid hole size, wherein the modified pattern of holes isderived from an initial patterns of holes.